Why does Maxwell's first equation make sense?

[yosimba2000]

So by definition, divergence means the ∫Flux Thru Differential Surfaces / ∫Differential Volume. So basically this just means we're calculating the fluxes divided by some factor, and Maxwell's first equation claims that ∇⋅E = q / ε_o is always true regardless of the shape of the surrounding surface.

I'm not understanding why this is true and how one could have figured this out. For a sphere it's clear that this is true, but how is it possible to generalize over all surfaces? I'm just thinking out loud here, but maybe all possible volumes are somehow the same as spheres, which is why they act the same? I realize that it's a law, so most likely it's just a postulate taken to be true, but is there a proof?

 

[PeroK]

So by definition, divergence means the ∫Flux Thru Differential Surfaces / ∫Differential Volume. So basically this just means we're calculating the fluxes divided by some factor, and Maxwell's first equation claims that ∇⋅E = q / ε_o is always true regardless of the shape of the surrounding surface.

I'm not understanding why this is true and how one could have figured this out. For a sphere it's clear that this is true, but how is it possible to generalize over all surfaces? I'm just thinking out loud here, but maybe all possible volumes are somehow the same as spheres, which is why they act the same? I realize that it's a law, so most likely it's just a postulate taken to be true, but is there a proof?

Try this:

https://en.wikipedia.org/wiki/Divergence_theorem

Or, any other resource on the Divergence Theorem.

 

[yosimba2000]

I'm not sure that helps. The divergence theorem is great to see how Gauss's Law comes about, but even more fundamental than Gauss's Law is Maxwell's 1st Equation in differential form, right? You need to accept Maxwell's 1st Equation in order to accept Gauss's Law. What's the proof/intuition on ∇⋅E = q / εo?

 

[Orodruin]

I'm not sure that helps. The divergence theorem is great to see how Gauss's Law comes about, but even more fundamental than Gauss's Law is Maxwell's 1st Equation in differential form, right? You need to accept Maxwell's 1st Equation in order to accept Gauss's Law. What's the proof/intuition on ∇⋅E = q / εo?

No, they are one and the same. Maxwell’s first equation is Gauss’s law on differential form. You can easily move from one form to the other by help of the divergence theorem.

 

[yosimba2000]

No, they are one and the same. Maxwell’s first equation is Gauss’s law on differential form. You can easily move from one form to the other by help of the divergence theorem.

Right, but the divergence theorem doesn't explain why the result of the differential form is true. As in, the divergence theorem shows ∫ (∇⋅E) dV = ∫ E⋅dS, but it doesn't show that (∇⋅E) = q / ε for any surface.

 

[Orodruin]

Right, but the divergence theorem doesn't explain why the result of the differential form is true.

It does if you assume the integral form. Just as assuming the differential form gives you the integral form.

As to the why, this is an empirical observation. Of course you can argue for it based on the field of a point charge, but this really is the wrong way around as what the field of a charge distribution is should be computed from Maxwell’s equations.

 

[yosimba2000]

But then the question becomes why Gauss's law is true for any surface? How is it that the flux through all surfaces, of any possible orientations, enclosing a volume, will always equal Q/ε_o? Maxwell just assumed this result?

Is this result just as fundamental as Newton's Laws?

 

[Orodruin]

But then the question becomes why Gauss's law is true for any surface?

You need to specify what you want to put as your basic assumption. Either it is Gauss's law on differential form or it is Gauss's law on integral form. Then you can show that the one you assumed directly results in the other. The inclusion of Gauss's law itself is a basic postulate of electromagnetism (in its typical formulation). Physics works by making a set of initial assumptions and then testing how well they predict how a system behaves. In the case of Maxwell's equations, they work very well for this purpose.

 

[yosimba2000]

You need to specify what you want to put as your basic assumption. Either it is Gauss's law on differential form or it is Gauss's law on integral form. Then you can show that the one you assumed directly results in the other. The inclusion of Gauss's law itself is a basic postulate of electromagnetism (in its typical formulation). Physics works by making a set of initial assumptions and then testing how well they predict how a system behaves. In the case of Maxwell's equations, they work very well for this purpose.

I see, thanks!